An Intermediate Value Theorem for the Arboricities

(1)

Volume 2011, Article ID 947151,7pages doi:10.1155/2011/947151

Research Article

An Intermediate Value Theorem for the Arboricities

Avapa Chantasartrassmee

1

_{and Narong Punnim}

2

1_{Department of Mathematics, School of Science, University of the Thai Chamber of Commerce,}

Bangkok 10400, Thailand

2_{Department of Mathematics, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, Thailand}

Correspondence should be addressed to Narong Punnim,narongp@swu.ac.th

Received 26 October 2010; Accepted 29 April 2011

Academic Editor: Aloys Krieg

Copyrightq2011 A. Chantasartrassmee and N. Punnim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

LetG be a graph. The vertexedge arboricity of Gdenoted by aGa1G is the minimum number of subsets into which the vertexedgeset ofGcan be partitioned so that each subset induces an acyclic subgraph. Let d be a graphical sequence and letRdbe the class of realizations of d. We prove that ifπ ∈ {a,a1}, then there exist integers xπ andyπ such that d has a

realizationGwithπG zif and only ifzis an integer satisfyingxπ ≤ z ≤ yπ. Thus, for an arbitrary graphical sequence d andπ ∈ {a,a1}, the two invariantsxπ minπ,d :

min{πG : G ∈ Rd}and yπ maxπ,d : max{πG : G ∈ Rd}naturally arise and henceπd :{πG :G∈ Rd} {z∈Z: xπ≤z ≤yπ}.We write drn _: _{r, r, . . . , r}

for the degree sequence of anr-regular graph of ordern. We prove thata1rn _{_r₁_/₂_}_.

We consider the corresponding extremal problem on vertex arboricity and obtain mina, rn_{in all}

situations and maxa, rn_{for all}_n_≥₂_r_2.

1. Introduction

We limit our discussion to graphs that are simple and finite. For the most part, our

notation and terminology follows that of Chartrand and Lesniak1. A typical problem in

graph theory deals with decomposition of a graph into various subgraphs possessing some prescribed property. There are ordinarily two problems of this type, one dealing with a decomposition of the vertex set and the other with a decomposition of the edge set. The vertex coloring problem is an example of vertex decomposition while the edge coloring problem

is an example of edge decomposition with some additional property. For a graph G, it is

always possible to partitionVGinto subsetsVi, 1≤i≤k, such that each induced subgraph

Vicontains no cycle. The vertex arboricity aGof a graphG is the minimum number of

subsets into whichVGcan be partitioned so that each subset induces an acyclic subgraph.

(2)

Chartrand et al. For a few classes of graphs, the vertex arboricity is easily determined. For

example, aCn 2. aKn n/2. Also aKr,s 1 if r 1 or s 1 andaKr,s 2

otherwise. It is clear that, for every graph G of order n, aG ≤ n/2. We now turn to

the second decomposition problem. The edge arboricity or simply the arboricity a1Gof a

nonempty graphGis the minimum number of subsets into whichEGcan be partitioned

so that each subset induces an acyclic subgraph. The arboricity was first defined by

Nash-Williams in3. As with vertex arboricity, a nonempty graph has arboricity 1 if and only if it

is a forest. Alsoa1Kn n/2.

A linear forest of a graphGis a forest ofG in which each component is a path. The

linear arboricity is the minimum number of subsets into whichEGcan be partitioned so that

each subset induces a linear forest. It was first introduced by Harary in4and is denoted by

ΞG. Note that the Greek letter, capital,Xi, looks like three paths! It was proved in5that

ΞG 2 for all cubic graphsGandΞG 3 for all 4-regular graphsG. It was conjectured

thatΞG r1/2for allr-regular graphsG.

A sequenced1, d2, . . . , dnof nonnegative integers is called a degree sequence of a graph

Gif the vertices ofGcan be labeledv1, v2, . . . , vnso that deg vi difor alli. If a sequence

d d1, d2, . . . , dnof nonnegative integers is a degree sequence of some graphG, then d is

called graphical sequence. In this case,G is called a realization of d. A graphGis regular of

degreerif deg vrfor each vertexvofG. Such graphs are calledr-regular. We write drn

for the sequencer, r, . . . , rof lengthn, whereris a nonnegative integer andnis a positive

integer. It is well known that anr-regular graph of ordernexists if and only ifn≥r1 and

nr ≡ 0mod 2. Furthermore, there exists a disconnectedr-regular graph of ordernif and

only ifn≥2r2.

LetGbe a graph andab, cd∈EGbe independent whereac, bd /∈EG. Put

Gσa,b;c,d _G_{− {}_{ab, cd}}_{{ac, bd}.} _1.1

The operationσa, b;c, dis called a switching operation. It is easy to see that the graph

obtained fromGby a switching has the same degree sequence asG.

Havel 6 and Hakimi 7 independently obtained that if G1 and G2 are any two

realizations of d, then G2 can be obtained fromG1 by a finite sequence of switchings. Let

Rddenote the set of all realizations of degree sequence d. As a consequence of this result,

Eggleton and Holton8defined, in 1978, the graphRdof realizations of d whose vertex set

is the setRd, two vertices being adjacent in the graphRdif one can be obtained from the

other by a switching. As a consequence of Havel and Hakimi, we have the following theorem.

Theorem 1.1. Let d be a graphical sequence. Then, the graphRdis connected.

Letπ∈ {a,a1}and d be a graphical sequence. Put

πd:{πG:G∈ Rd}. 1.2

A graph parameterπis said to satisfy an intermediate value theorem over a class of graphs

JifG, H ∈ JwithπG< πH, then, for every integerkwithπG≤ k ≤πH, there is

a graphK ∈ Jsuch thatπK k. If a graph parameterπ satisfies an intermediate value

theorem overJ, then we writeπ,J∈IVT. The main purpose of this section is to prove that

(3)

Theorem 1.2. Let d be a graphical sequence andπ ∈ {a,a1}. Then, there exist integersxπand

yπsuch that there exists a graphG∈ RdwithπG zif and only ifzis an integer satisfying

xπ≤z≤yπ. That is,πd {z∈Z:xπ≤z≤yπ}.

The proof of Theorem1.2follows from Theorem1.1and the following results.

Lemma 1.3. Letπ ∈ {a,a1}andσbe a switching on a graphG. Then,πGσ≤πG 1.

Proof. Let G be a graph andσ σa, b;c, d be a switching on G. Then, aGσ _{− {a, b}} _≤

aGand henceaGσ _≤ _aGσ_{− {a, b}}₁ _≤ _aG_{1. Since}_{ac, bd} _∈ _EGσ_{, we have that}

a1Gσ− {ac, bd}≤a1Gand hencea1Gσ≤a1Gσ− {ac, bd} 1≤a1G 1.

Letπ ∈ {a, a1}. By Lemma1.3, ifπG≤πGσ, then we have thatπG ≤πGσ ≤

πG 1. IfπG≥πGσ_{, then, by Lemma}_{1.3, we see that}_πG_πGσσ−1

≤πGσ ₁_≤

πG 1, whereσ−1σa, c;b, d.

We have the following corollary.

Corollary 1.4. Let G be a graph and let σ be a switching on G. If π ∈ {a,a1}, then |πG−

πGσ_{| ≤}_1.

2. Extremal Results

Letπ ∈ {a,a1}. By Theorem1.2,πdis uniquely determined byxπandyπ, thus it is

reasonable to denotexπ minπ,d : min{πG : G ∈ Rd}andyπ maxπ,d :

max{πG : G ∈ Rd}. We first state a famous result on edge arboricity of Nash-Williams

3.

Theorem 2.1. For every nonempty graph G, a1G max|EH|/|VH| −1, where the

maximum is taken over all nontrivial induced subgraphsHofG.

As a consequence of the theorem, it follows thata1Kn n/2anda1Kr,s rs/r

s−1.

Ifπ ∈ {a,a1}andHis a subgraph ofG, thenπH ≤ πG. It is well known that if

Gis a graph with a degree sequence d d1, d2, . . . , dn d1 ≥ d2 ≥ · · · ≥ dn, then Gcan

be embedded as an induced subgraph of ad1-regular graphHandπG≤πH. Thus, it is

reasonable to determine minπ,dand maxπ,din the case where drn_{. It should be noted}

that ifr ∈ {0,1}, thena1G 1 for allG ∈ Rrn. Therefore, we may assume from now on

thatr≥2.

Theorem 2.2. mina1, rn maxa1, rn r1/2.

Proof. LetGbe anr-regular graph of ordern. By Theorem2.1, we see thata1G≥ nr/2n−

1for anyr-regular graphGof ordern. Sincenr/2n−1 r/2r/2n−1, it follows

thata1G≥ r1/2. LetHbe an induced subgraph ofGof orderm. Ifm ≥r 1, then

|EH|/|VH|−1≤mr/2m−1. It follows that|EH|/|VH|−1≤r1/2. Ifm≤r,

then|EH| ≤m₂. It is clear that|EH|/|VH| −1 ≤r1/2. Thus,a1G≤r1/2.

Therefore, mina1, rn maxa1, rn r1/2.

(4)

LetX and Y be finite nonempty sets. We denote by KX, Ythe complete bipartite

graph with partite setsXandY. LetGandHbe any two graphs. Then,G∗His the graph

withVG∗H VG∪VHandEG∗H EG∪EH. It is clear that the binary operation

“∗” is associative. IfG1andG2are disjointi.e.,VG1∩VG2 ∅, thenG1∗G2 G1∪G2.

We also usepGfor the union ofpdisjoint copies ofG.

It should be noted once again that anr-regular graphG of ordernhaveaG 1 if

and only ifr ∈ {0,1}. Thus, mina, rn _{maxa, r}n _{1 if and only if}_r _{∈ {0,}_{1}. Moreover,}

mina,2n maxa,2n 2.

Letrandnbe integers withr≥3 andRrn_/_{∅. Then,}

1ifnis even andn≥ 2r, then there exists anr-regular bipartite graphGof ordern.

Therefore,aG 2;

2ifnis odd andn≥2r1, thenris even and there exists anr-regular bipartite graph

H of ordern−1. LetFbe a set of r/2 independent edges ofH. Then,H−F is a

bipartite graph of ordernhavingrvertices of degreer−1 andn−1−rvertices of

degreer. LetGbe a graph obtained fromH−Fby joining thervertices ofH−Fto

a new vertex. Therefore,Gis anr-regular graph of ordernhavingaG 2;

3if n is even and n 2r − 1, then Kr−1,r−1 is the complete r − 1-regular

bipartite graph of order n. Let X {x1, x2, . . . , xr−1} and Y {y1, y2, . . . , yr−1}

be the corresponding partite sets ofKr−1,r−1. LetG be defined byG Kr−1,r−1 −

{x2y2, x3y3, . . . , xr−2yr−2} {x1x2, x2x3, . . . , xr−2xr−1, y1y2, y2y3, . . . , yr−2yr−1}. Then

Gis anr-regular graph of ordernhavingaG 2;

4if n 2r − 1, then r is even. Let X and Y be as described in the

previous case and u be a new vertex. Then, H KX, Y E1, where E1

{ux1, uy1, ux2, uy2, . . . , uxr/2, uyr/2}, is a graph of order 2r−1 havingr1 vertices

of degreerandr−2 vertices of degreer−1. PutX1{xi∈X:r/2< i≤r−1}and

Y1{yi∈Y :r/2< i≤r−1}. Thus,|X1||Y1| r−2/2.

Case 1. Ifr−2/2 is even, then we can construct anr-regular graphGfromHby adding

r−2/4 independent edges to each ofX1andY1. This construction yieldsaG 2.

Case 2. Ifr−2/2 is odd, thenH−xr/2yr/2is a graph havingr−1 vertices of degreerandr

vertices of degreer−1. Thus, it hasr/2 vertices of degreer−1 inXand alsor/2 vertices inY.

Sincer/2 is even, we can easily construct anr-regular graphGfromH−xr/2yr/2by adding

appropriate independent edges so thataG 2.

With these observations, we easily obtain the following result.

Lemma 2.3. mina, rn _{2 for all}_n_≥_2r₋_2.

In order to obtain mina, rn _{in all other cases, we first state some known result}

concerning the forest number of regular graphs. LetGbe a graph andF ⊆VG.F is called

an induced forest ofGif the subgraphFofGcontains no cycle. The maximum cardinality

of an induced forest of a graphGis called the forest number ofGand is denoted byfG. That

is,

(5)

The second author proved in 9 the following result: Letr and s be integers with

1≤s≤r. Then,

1fG≤s1 for allG∈ Rrrs_,

2there exists a graphH∈ Rrrs_{such that}_fH_s_1.

With these observations, we have the following result.

Lemma 2.4. If 1≤s≤r−3, then mina, rrs≥ rs/s1.

It is well known thataKr1 r1/2. Thus, mina, rr1 maxa, rr1 r

1/2. It is also well known thatG Kr2−F, whereF is a 1-factor ofKr2, is the unique

r-regular graph of orderr2. SinceEG F, it follows that a subgraph ofGinduced by

any four vertices ofVGcontains at most two edges. Equivalently, a subgraph ofGinduced

by any four vertices must contain a cycle. Thus,aG ≥ r2/3. It is easy to show that

aG r2/3. Thus, mina, rr2 _{maxa, r}r2 _r_2/3.

LetTn,qbe the Tur´an graph of orderncontaining no clique of orderq1. Thus, there

exists a unique pair of integersxandy such thatn qxy, 0 ≤ y < q. Note thatTn,q is

a completeq-partite graph of ordernconsisting ofypartite sets of cardinalityx1 andq−y

partite sets of cardinalityx. Therefore,Tn,qis ann−x-regular graph if and only ify0. If

y >0, thenTn,qcontainsyx1vertices of degreen−x−1 andq−yxvertices of degree

n−x. A little modification ofTn,qyields a regular graph with prescribed arboricity number as

in the following theorem.

Theorem 2.5. Letn, x, q, andybe positive integers satisfyingnqxy, 0< y < q. Then,

1there exists ann−x-regular graphGof ordernsuch thataG≤qifxis odd,

2there exists ann−x-regular graphGof ordernsuch thataG≤qifxandyare even,

3there exists ann−x−1-regular graphHof ordernsuch thataG≤qifxis even and

yis odd.

Proof. Note thatTn,q is a completeq-partite graph of ordernconsisting ofy partite sets of

cardinalityx1 andq−ypartite sets of cardinalityx. LetV1, V2, . . . , Vybe partite sets ofTn,q

of cardinalityx1 andU1, U2, . . . , Uq−ybe partite sets ofTn,qof cardinalityx. It is clear that

deg_T_n,q vn−x−1 ifv∈V1∪V2∪ · · · ∪Vyand deg_T_n,q un−xifu∈U1∪U2∪ · · · ∪Uq−y.

For eachi 1,2, . . . y, letVi {vi1, vi2, . . . , vix1}; similarly, for eachj 1,2, . . . , q−y, let

Uj{uj1, uj2, . . . , ujx}.

1Suppose thatxis odd. Since|Vi| x1 for eachi 1,2, . . . , y, ann−x-regular

graphGcan be obtained fromTn,qby addingx1/2 independent edges to eachVi, 1≤i≤y.

It is clear thataG≤q.

2Suppose thatxandyare even. LetHy_i₁PViwherePViis a path withVias

its vertex set andEi{vikvik1: 1≤k≤x}as its edge set. LetF1, F2, . . . , Fy/2be defined by

Fk{vktvky/2t: 2≤t≤x}and putF

y/2

k1Fk. Finally, we obtain a graphG Tn,q−F∗H

which is ann−x-regular graph of ordernandaG≤q.

3Suppose thatxis even andyis odd. Then,nqxyis odd. Ifq−y≥2, then, by

Dirac10, a subgraph ofTn,qinduced byU1∪U2∪ · · · ∪Uq−ycontains a Hamiltonian cycle

C. Sincexis even andCis of orderxq−y, it follows thatCis of even order. LetFbe a set

ofxq−y/2 independent edges ofC. Then,H Tn,q−Fis ann−x−1-regular graph of

(6)

Suppose that q− y 1. Let F1 {v11u11, v12u12, . . . , v1xu1x} and F2 {v11v12,

v13v14, . . . , v1x−1v1x}. Then,H Tn,q−F1 F2 is ann−x−1-regular graph of ordern

andaH≤q.

Note that ifGis anr-regular graph onrsvertices where 1≤s≤r, thenr≥rs/2.

It follows, by Dirac10, thatGis Hamiltonian.

Theorem 2.6. Letrandsbe integers withr≥3 and 1≤s≤r−3. Then,

mina, rrs

rs

s1

. 2.2

Proof. By Lemma2.4, we have that mina, rrs_{≥ r}_s/s₁_{if 1} _≤_s _≤ _r₋_{3. We have}

already shown that mina, rrs _r _s/s₁ _if _s _{∈ {1,}_{2}. Thus, in order to prove}

this theorem, it suﬃces to construct anr-regular graphG of orderrs such thataG ≤

rs/s1forr≥3 and 3≤s≤r−3. Suppose that 3≤s≤r−3. Letqrs/s1

and putrsqxy, 0≤y≤q−1. Then,x≤s1 andxs1 if and only ify0 andrs

is a multiple ofs1.

Note thatTn,q is a completeq-partite graph of ordernconsisting ofqpartite sets of

cardinalitys1. Suppose thatsis odd. LetGbe a graph obtained fromTn,qby addings1/2

independent edges to each partite set. Then,Gis anr-regular graph of ordernhavingaG

q.

Ifsis even, thenqis even. By the same argument with 2 in Theorem2.5, there exists

anr-regular graphGobtained fromTn,qwithaG q.

Now suppose thaty > 0 and thereforex ≤ s. Thus,Trs,q containsy partite sets of

cardinalityx1 andq−ypartite sets of cardinalityx. By Theorem2.5, there exists anr

s−x-regular graphGof orderrswithaG ≤qifxis odd or bothxandyare even and there

exists anrs−x−1-regular graphHof orderrswithaH≤qifxis even andyis odd.

Sincex≤s, it follows thatrs−x≥r. We will consider two cases according to the parity of

xandyas in Theorem2.5.

Case 1. Ifxis odd or bothxandyare even, then, by Theorem2.5, there exists anrs−

x-regular graphG of orderr swith aG ≤ q. Sincex ≤ s, it follows that rs−x ≥ r. If

s−x0, thenG∈ Rrrs_with_aG_≤_{q, as required. Suppose that}_s₋_x_{is even and}_s₋_x_≥_2.

By Dirac10,Gcontains enough edge-disjoint Hamiltonian cycles whose removal produces

anr-regular graphGof orderrsandaG≤aG≤q, as required.

Supposes−xis odd. Then, eitherrs−xorris odd and thereforersis even. Also

by Dirac10,G contains a 1-factorF whereG−F is anr s−x−1-regular graph and

aG−F≤aG≤q. Sinces−x−1 is even, the result follows by the same argument as above.

Case 2. Ifxis even andyis odd, then, by Theorem2.5, there exists anrs−x−1-regular

graphHof orderrsandaH≤q. Sincersqxyis odd, it follows that bothrs−x−1

andr are even. Sincex ≤s, it follows thatrs−x−1 ≥ r−1. Therefore, it follows by the

parity thatrs−x−1≥r. The result follows easily by the same argument as in Case1.

This completes the proof.

Chartrand and Kronk11obtained a good upper bound for the vertex arboricity. In

(7)

Theorem 2.7. For each graphG,aG≤1maxδG/2, where the maximum is taking over all

induced subgraphsGofG. In particular,aG≤1r/2ifGis anr-regular graph.

In general, the bound given in Theorem2.7is not sharp but it is sharp in the class of

r-regular graphs of ordern≥2r2 as we will prove in the next theorem.

Theorem 2.8. Forn≥2r2,maxa, rn ₁_r/2.

Proof. The result follows easily ifr∈ {0,1,2}. Suppose thatr≥3. We writen r1qt, 0≤

t≤ r. Thus,q≥2. Note that anr-regular graph of ordernexists implyingrnmust be even.

Thus, ifris odd, thennmust be even and hencetmust be even. PutG q−1Kr1∪H,

whereHis anr-regular graph of orderrt1. SinceaKr1 r1/21r/2and

aH≤1r/2, it follows thataG 1r/2.

Acknowledgments

The first author would like to thank University of the Thai Chamber of Commerce for financial support. The second author gratefully acknowledges the financial support provided by the Thailand Research Fund through the Grant number BRG5280015. The authors wish to thank the referee for useful suggestions and comments that led to an improvement of this paper.

References

1 G. Chartrand and L. Lesniak, Graphs & Digraphs, Chapman & Hall/CRC, London, UK, 4th edition, 2005.

2 G. Chartrand, D. Geller, and S. Hedetniemi, “Graphs with forbidden subgraphs,” Journal of

Combina-torial Theory, vol. 10, pp. 12–41, 1971.

3 St. J. A. Nash-Williams, “Decomposition of finite graphs into forests,” Journal of the London

Mathemat-ical Society, vol. 39, p. 12, 1964.

4 F. Harary, “Covering and packing in graphs. I,” Annals of the New York Academy of Sciences, vol. 175, pp. 198–205, 1970.

5 J. Akiyama, G. Exoo, and F. Harary, “Covering and packing in graphs. III: cyclic and acyclic invari-ants,” Mathematica Slovaca, vol. 30, no. 4, pp. 405–417, 1980.

6 V. Havel, “A remark on the existence of finite graphsCzech,” ˇCasopispest Mathematical Journal, vol.

80, pp. 477–480, 1955.

7 S. L. Hakimi, “On realizability of a set of integers as degrees of the vertices of a linear graph,” SIAM

Journal on Applied Mathematics, vol. 10, pp. 496–506, 1962.

8 R. B. Eggleton and D. A. Holton, “Graphic sequences, combinatorial mathematics, VI,” in Proceedings

of the 6th Australian Conference, vol. 748 of Lecture Notes in Math., pp. 1–10, University of New England,

Armidale, Australia, 1978.

9 N. Punnim, “Decycling regular graphs,” The Australasian Journal of Combinatorics, vol. 32, pp. 147–162, 2005.

10 G. A. Dirac, “Some theorems on abstract graphs,” Proceedings of the London Mathematical Society, vol. 2, pp. 69–81, 1952.

11 G. Chartrand and H. V. Kronk, “The point-arboricity of planar graphs,” Journal of the London

(8)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

International Journal of Hindawi Publishing Corporation

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

Journal of

An Intermediate Value Theorem for the Arboricities

Research Article